#10094642

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,4,7)(3,6,8,5)(17,18)(19,24)(20,22)(21,28)(23,27)(25,30)(26,31)(29,32);(3,8)(5,6)(9,10,13,16)(11,12,15,14)(17,19,22,27)(18,23,20,24)(21,26,30,29)(25,31,28,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,16)(13,15)(18,20)(23,24)(25,28)(31,32);(3,8)(5,6)(9,12)(10,15)(11,16)(13,14)(17,20,22,18)(19,24,27,23)(21,28,30,25)(26,32,29,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,15)(14,16)(17,19)(18,24)(20,23)(21,29)(22,27)(25,31)(26,30)(28,32);(1,2)(3,5)(4,7)(6,8)(9,12)(10,15)(11,16)(13,14)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10001838);